Patterns are everywhere in the animal kingdom but understanding the mechanisms that produce them is a real challenge.
In this week's Physical Review E Thomas Woolley and Ruth Baker of Oxford University's Mathematical Institute report on mathematical simulations that may explain how stingrays generate their distinctive spots.
I asked Thomas about this work and the relationship between maths and nature's spots and stripes:
OxSciBlog: What are the challenges in reproducing these patterns?
Thomas Woolley: The biology! Mathematically, we have a number of models that produce qualitatively the same patterns as animal skins. However, there are currently no specific biological examples which can be unequivocally linked to the maths.
The specific difficulty with stingrays, which is why they have not been considered before, is that their spots have a dark halo around the central spot. The BVAM model is one of the first biological pattern formation systems that is able to produce this dark halo.
The Barrio-Aragon-Varea-Maini (BVAM) model is a set of mathematical equations that models two chemicals which are able to diffuse and react in a domain. They are generalisations of all other such pattern forming systems and therefore very complex. They are used as a way explore a wide variety of chemical dynamics from two simple equations.
OSB: How did you set out to reproduce stingray patterns?
TW: It was in fact the other way around. We were able to produce the patterns and thus wanted to find a biological example that exemplified them.
As mentioned above, there are a great number of mathematical models which can produce animal pigmentation patterns. A particularly large group of such models are called Turing systems (named after Alan Turing who originally considered them).
Normally, these models give only a single particular type of pattern; either spots or stripes. The BVAM model, we considered, is a generalisation of all of these. This means that it can give many different types patterns (see below). It was noticed that the spots that the model produces were not of the normal Turing type (Turing type spots do not have the dark halo).
In a precursor paper it was shown that these are very similar to a stingray’s skin pattern. We generalised the results that appeared in the previous paper and showed that the spots can exist for a larger range of parameters than previously thought.
OSB: How could your results be tested?
TW: The analysis we have done on the model connects the parameters of the equations and the size of the spots. An initial test would simply be to measure the spots on a large number of stingrays. By varying the model parameters we can vary the size of the simulated spots, however, if we are unable to produce the correct size, which correspond to reality we immediately show that the BVAM is not the system behind stingray patterning.
The best way would be to try and discover the stingrays’ chemical signals which produce the spots and, either, show it corresponds to, or differs from, the BVAM equations. This is currently a difficult problem for biologists, which is why there are currently no clear biological examples. In the end, biologists will probably never be able to “prove” a model is correct only that it fits current data and is thus not incorrect.
Hence, if biologists are able to alter the system to produce a pattern that the BVAM system cannot reproduce we must conclude that the BVAM system is not biologically accurate and return to the drawing board. However, a disproved model is still important as it implies what adaptations are needed in order to generate a more refined system.
OSB: What do these results tell us about biological mechanisms?
TW: The patterning systems we use tend to rely on diffusion as the key mechanism. In terms of evolution this is important as it suggests that no energy from the animal is needed to produce the pattern; only to create the chemicals which will naturally diffuse.
Another important aspect is that it suggests many types of fish depend on the exact same model to produce their individual patterns. This supports the idea that evolution has simply picked the simplest mechanism, whilst mutations and various types of selection will specify how the model behaves.
OSB: How could what you learned be applied to other problems?
TW: The BVAM system, because of its general nature, not only has applications in animal skin patterning, but it has also been linked to a model of cardiac regulation and, further, it has the potential to be used in encoding digital information (the spots can act like binary digits).
The particular use of my work will be the analytical methods that we have produced, which can be used on many similar problems in various fields. Further, it broadens the number of patterns which can be treated mathematically. For a long time we have been able to consider spots and stripes, but we are the first to consider the biological applications of the dark-ringed spot.
Thomas Woolley is based at Oxford University's Mathematical Institute.